Question

Which of the following expresses ${\left(\frac{x^2\cdot y^4}{z^3}\right)}^0$ in terms of $p$ ? (A) 0 (B) 1 (C) $p^{-5}$ (D) $p$

Short answer

The correct answer is (B) 1, as ${\left(\frac{x^2\cdot y^4}{z^3}\right)}^0=1$ according to the rule of exponents, where any non-zero real number raised to the power of 0 is equal to 1.

Step-by-step solution

Apply the Rule of Exponents

Given ${\left(\frac{x^2\cdot y^4}{z^3}\right)}^0$, we will apply the rule of exponents: any non-zero real number raised to the power of 0 is equal to 1. So, we will have: $${\left(\frac{x^2\cdot y^4}{z^3}\right)}^0=1$$

Select the correct option

Now that we have simplified the expression, we will look at the provided options. The correct answer is the one that matches the simplified expression: (A) 0 - No, this is not equal to 1. (B) 1 - Yes, this matches our simplified expression. (C) $p^{-5}$ - No, this is not equal to 1. (D) $p$ - No, this is not equal to 1. Therefore, the correct answer is (B) 1.

Key concepts

Exponents and Powers

Understanding exponents and powers is crucial for mastering various topics in mathematics. In essence, an exponent tells us how many times a number, known as the base, is multiplied by itself. For example, in the expression $3^4$, 3 is the base and 4 is the exponent, indicating that 3 is multiplied by itself 4 times, which equals to $3\times 3\times 3\times 3=81$.

One of the most fundamental rules involving exponents is dealing with a base raised to the zero power. Regardless of the base (as long as it is not zero), the result of any number raised to the power of zero is always 1. This rule is applied in our textbook solution, where we encounter $(\frac{x^2\cdot y^4}{z^3}{)}^0$. According to the zero exponent rule, this simplifies directly to 1, an essential concept that often appears in algebra and beyond.

Additionally, understanding the power of powers can simplify complex equations, especially when variables are involved. For instance, raising a power to another power, as seen in expressions like $(x^m{)}^n$, results in the powers being multiplied, yielding $x^{mn}$. These fundamental rules are the keystones of working with exponents in higher-level mathematics.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (such as $x,y,$ and $z$), and operation signs. Variables represent unknown values and are a core component of algebra. Math problems often require us to manipulate these expressions through various operations to solve for the unknowns or to simplify the expression.

When dealing with algebraic expressions, it's important to understand the terms and how they combine. For example, in $x^2\cdot y^4$, $x^2$ and $y^4$ are individual terms that are being multiplied together. Algebraic expressions can be simplified, expanded, or factored, and each of these operations uses a different set of rules or properties of algebra.

An interesting aspect of algebraic expressions is their ability to encode relationships between quantities. They can range from very simple, like $x+5$, to much more complex forms, like the one in our exercise. As students advance in their studies, they'll encounter algebraic expressions in topics like calculus, statistics, and even in real-world applications such as computing and engineering.

Simplifying Expressions

Simplifying expressions is the process of reducing an algebraic expression to its simplest form. This means combining like terms, applying exponent rules, and performing any possible arithmetic to make the expression as readable and as straightforward as possible. Simplification is not just about making the expressions shorter; it often makes other operations, such as solving equations or evaluating expressions, much easier.

In the context of our exercise, the simplification was straightforward due to the presence of an exponent of zero. With a more complex expression, simplification would involve additional steps such as distributing multiplication over addition $a(b+c)=ab+ac$, combining like terms, and reducing fractions. It's a powerful tool in algebra that's used frequently to solve equations efficiently.

A common mistake is to overlook the rules and laws of algebra when simplifying expressions, which can lead to incorrect solutions. Clear and consistent application of these rules ensures that students can accurately simplify expressions and further their understanding of algebraic principles.